{ -2 x+2 y+3 z =c x-y =d y+4 z =e. (a) [ c d e ]=[ 1 3

{ -2 x+2 y+3 z =c x-y =d y+4 z =e. (a) [ ...

Key Concepts

Parameter Dependency in Linear Systems

Parameter dependency refers to the presence of additional parameters (or unknown constants) within a system of linear equations that can affect the existence and uniqueness of solutions. Analyzing how particular choices or conditions on these parameters influence the consistency and solution set of the system is a key aspect of studying linear systems, especially when the parameters are part of the coefficients or constants.

Consistency and Inconsistency of Systems

In the context of linear systems, a system is said to be consistent if it has at least one solution (which may be unique or infinite in number), and inconsistent if it has no solutions. Determining consistency often involves checking if the augmented matrix has any contradictory rows after row reduction, which implies that no set of variable values can satisfy all the equations simultaneously.

Augmented Matrix Representation

An augmented matrix is a compact way to represent a system of linear equations by including the coefficients of the variables as well as the constants from the right-hand side of the equations. This representation is particularly useful for applying algorithmic methods, such as Gaussian elimination or row reduction, to solve the system efficiently.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Solving such systems involves finding the values of the variables that satisfy all of the equations simultaneously. This concept is central to algebra and has applications in various fields such as engineering, economics, and physics.

Transcript

00:01 In the question given system of equations are that is minus 2x plus 2y plus 3 z is equal to c, x minus y is equal to d and y plus 4 z is equal to e.

00:20 Now in the question it is asking to find the solution of this system by using inverse method.

00:29 So we have to apply the inverse method for finding the solution.

00:34 So first as you see in the question there are two parts given.

00:39 So in the part a it is saying that the matrix of cd is equal to matrix of 1 3 minus 2.

00:50 And first i will find the value, find the answer for part a, then after i will find the answer for next part that is part b so for finding the solution first i will write the matrix equation that is equal to that is written as a x is equal to b or i can say that x will be equal to a inverse b here a is a here x is equal to a inverse we become by after multiplying a inverse on both side of this first equation so then i will get that is x is equal to a inverse b now here matrix a is a coefficient of x y it means matrix a will be written as that is bracket and in the elements are that is minus 2 xyz so means coefficient of xyz so it will be written as minus 2 2 3 here i will write 1 minus 1 and 0 because the coefficient of z is 0 and here i will write 0 because coefficient of x is 0 for third equation that is given and here that is 1 and here i will write 4 so this is the matrix a and matrix x is a matrix of x y and matrix b is a matrix of cde whose value given that is 1 3 minus 2 now next step is we have to find the inverse of matrix a so after swalling i will get the value of matrix a that is equal to value is 1 divided by 3 and in the bracket the elements are that is minus 4 minus 5 3 minus 4 minus 8 3 1 to 0 so this is the value of a inverse and now use the equation matrix equation that is x is equal to a inverse b so it will be written as matrix of x y is equal to a inverse a inverse which i written right in the the us step that is 1 divided by 3 and the bracket the elements are minus 4 minus 5 3 minus 4 minus 8 3 and this will be this row elements that is 1 to 0 multiplied by matrix b matrix b is that is 1 3 minus 2 now next step is we have to multiply we have to multiply this both matrix by using dot product so after multiplying i will get the elements in the matrix that is equal to 1 divided by 3 and 4 first matrix sorry first elements by using the dot product what we have to do is we have to multiply first row with the column means this first row with this column so i can say with the matching elements so it will be written as minus four times of one then minus five times of three then three times of minus two now for second element we have to multiply second row with column means this second row and this column with the matching elements so it will be written as minus four times of one then minus 8 times of 3 and then plus 3 times of minus 2...

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